I wouldn't use the appearance method on this one, since it's rather difficult to get an exact result. I would rather create circles with 6 anchors (this is possible with the script by s.h.). Those cricles can then scnap exactly. After that I would also be careful with the shapebuilder tool since it might produce unwanted result (especially closing of paths you don't want closed). Instead direct select the segments you dont need and just delete them.

Here is a quite different way, silly maybe, but completeæly accurate and very simple, especially if you just let it grow without deleting (much) underway. If you remove the outer circle, you may just click and delete top duplicates in the different stages where deletion is mentioned (Undo is your friend when you overdo it).

You may add the large circle as a copy (3 diameters) at any time if desired.

eidt: How come my inserted image got scaled down?. I thought the forum scales down only images wider than 450 pixels mine is 445. What's the trick? I know clicking on it will show it in full size but I was hoping to avoid that extra step.